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10.1146/annurev.biophys.33.110502.132716
Annu. Rev. Biophys. Biomol. Struct. 2004. 33:245–68
doi: 10.1146/annurev.biophys.33.110502.132716
c 2004 by Annual Reviews. All rights reserved
Copyright °
First published online as a Review in Advance on January 16, 2004
ROTATION OF F1-ATPASE: How an ATP-Driven
Molecular Machine May Work
Kazuhiko Kinosita, Jr.,1 Kengo Adachi,1 andHiroyasu Itoh2,3
1
Center for Integrative Bioscience, Okazaki National Research Institutes, Higashiyama
5-1, Myodaiji, Okazaki 444-8585, Japan; email: [email protected]; [email protected]
2
Tsukuba Research Laboratory, Hamamatsu Photonics KK, and 3CREST “Creation and
Application of Soft Nano-Machine, the Hyperfunctional Molecular Machine” Team 13∗ ,
Tokodai, Tsukuba 300-2635, Japan; email: [email protected]
Key Words molecular motor, single-molecule physiology, torque generation,
efficiency of energy conversion, free energy
■ Abstract F1-ATPase is a rotary motor made of a single protein molecule. Its
rotation is driven by free energy obtained by ATP hydrolysis. In vivo, another motor,
Fo, presumably rotates the F1 motor in the reverse direction, reversing also the chemical
reaction in F1 to let it synthesize ATP. Here we attempt to answer two related questions,
How is free energy obtained by ATP hydrolysis converted to the mechanical work of
rotation, and how is mechanical work done on F1 converted to free energy to produce
ATP? After summarizing single-molecule observations of F1 rotation, we introduce a
toy model and discuss its free-energy diagrams to possibly answer the above questions.
We also discuss the efficiency of molecular motors in general.
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F1 -ATPase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ROTATION OF F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Single-Molecule Observations of F1 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Potential Energy for γ Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How an ATP-Driven Molecular Machine May Work . . . . . . . . . . . . . . . . . . . . . . . .
A CAMSHAFT MODEL OF F1 ROTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Push-Pull Mechanism for Torque Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Mechanism Warranting Unidirectional Rotation . . . . . . . . . . . . . . . . . . . . . . . .
Clockwise Rotation Leads to ATP Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
POTENTIAL DIAGRAMS FOR THE CAMSHAFT MODEL . . . . . . . . . . . . . . . . . .
Free-Energy Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Major Reaction Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Other Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EFFICIENCY OF CHEMO-MECHANICAL ENERGY CONVERSION . . . . . . . . .
1056-8700/04/0609-0245$14.00
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Responses to Decreasing 1G in the Absence of External Load . . . . . . . . . . . . . . . .
Rotation Against a Conservative Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dissipation and Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HOW TO DESIGN AN ATP-DRIVEN MOLECULAR MACHINE . . . . . . . . . . . . . .
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INTRODUCTION
F1-ATPase
F1-ATPase is a part of the enzyme ATP synthase that synthesizes ATP from ADP
and inorganic phosphate (Pi) using proton flow across the membrane as the energy
source (10, 26, 60, 67). The membrane-embedded Fo portion conducts protons,
which drive the catalytic synthesis of ATP in the water-soluble F1 portion. The
ATP synthase is a reversible molecular machine in that hydrolysis of ATP in F1
can pump protons through Fo in the opposite direction against an electrochemical
gradient of protons. To explain the energetic coupling between the proton flow in
Fo and synthesis/hydrolysis of ATP in F1, Boyer & Kohlbrenner (12) and Oosawa
and colleagues (17, 43) independently proposed rotational catalysis at about the
same time, although a full account of the latter’s model (44) appeared late. The
prediction was that both Fo and F1 are rotary motors, Fo being powered by proton
flow and F1 by ATP hydrolysis, and that the genuine rotary directions of the two
are opposite to each other yet the two have a common rotary shaft (rotor). When
the drop of proton electrochemical potential across the membrane exceeds the
free energy obtained by ATP hydrolysis, Fo wins and F1 is forced to rotate in its
reverse direction, resulting in the synthesis of ATP in its catalytic sites. If the free
energy of ATP hydrolysis is larger, rotation is in the direction of F1 and protons are
pumped back. Crystal structures of F1 (1) and FoF1 (53) suggested that the two are
indeed rotary motors, and unidirectional rotation, counterclockwise when viewed
from the Fo side, was videotaped for an isolated F1 attached to a glass surface (40).
For the whole ATP synthase, reorientation of γ under proton-motive force has
been indicated in a cross-linking study (68), and single-molecule study (9a) has
suggested rotation. Proton driven rotation, clockwise and stepwise, has recently
been demonstrated in a fluorescence resonance energy transfer study (70).
Isolated F1, consisting of α 3β 3γ δε subunits (in bacteria), only hydrolyzes ATP
and hence is called F1-ATPase. An α 3β 3γ subcomplex suffices for rotation and
hydrolysis activities: The central γ subunit rotates inside a cylinder made of three
α and three β subunits arranged alternately (Figures 1 and 2) when ATP is hydrolyzed in three catalytic sites each hosted primarily by a β subunit. In this
article, we focus on this simplest subcomplex and, hereafter, refer to it simply
as F1.
The Question
The central question we ask in this article is, How are the chemical reactions,
synthesis or hydrolysis of ATP, in the catalytic sites coupled to the mechanical
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rotation of γ ? Hydrolysis of one molecule of ATP accompanies a free-energy drop
1G of
1G = 1G 0 + k B T ln[ADP][Pi]/[ATP],
1.
where 1G0 = −50 pN · nm is the standard free-energy change at pH 7 (6; also
see recent beautiful work in Reference 56), and kBT = 4.1 pN · nm is the thermal
energy at room temperature (kB, the Boltzmann constant; T, absolute temperature).
Under physiological conditions of [ATP] ∼10−3 M, [ADP] ∼10−4 M, and [Pi]
∼10−3 M, 1G is about −90 pN · nm. 1G is negative under all practical conditions,
implying that the reaction always proceeds in the direction of hydrolysis and that
spontaneous net synthesis of ATP never happens in solution. How, then, is this
chemical reaction, which is destined to proceed in one direction, converted to
unidirectional rotation that can do work against an external opposing torque?
In this article, we first summarize rotational characteristics of F1 revealed by
single-molecule observations. The results allow us to deduce potential energies for
γ rotation in various chemical states. We then try to answer the central question
on the basis of these potential energies. In doing so, we introduce a toy model
that can reproduce the experimental potential energies and that, if fabricated by
future nanotechnology, would work as an ATP-driven rotary motor. Our goal is to
show a possible, concrete mechanism of chemo-mechanical energy transduction
that bears some essential features of the actual F1 motor. We discuss how a motor
senses free energy of ATP hydrolysis and how it responds to an external load. Our
discussion is yet premature and may contain incorrect statements, but we hope that
some of the concepts we present here may be useful in interpreting or designing
future experiments.
ROTATION OF F1
Single-Molecule Observations of F1 Rotation
Counterclockwise rotation was first imaged in F1 from a thermophilic bacterium by
attaching a fluorescent actin filament to γ while fixing the α 3β 3 cylinder to a glass
surface (40). Since then, counterclockwise rotation has been confirmed in F1 from
Escherichia coli (39, 41) and plants (19). Rotation of tags other than actin has been
imaged successfully (2), including a single fluorophore (2a), spherical bead or bead
duplex (18, 65), metal bar (52), or a single donor-acceptor pair for fluorescence
resonance energy transfer (9, 63). Subunits attached to γ rotate counterclockwise
during ATP hydrolysis, suggesting that they are part of the common rotor shaft in
the whole ATP synthase: the ε subunit (24) and the ring of c subunits in Fo (37, 49,
55), but not the a subunit of Fo (37), which is considered to be part of the stator.
V1-ATPase, a relative of F1-ATPase, also rotates in the same direction (21).
Below we summarize the rotational characteristics of the F1 motor that are
essential to the understanding of the mechanism of chemo-mechanical coupling.
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Most of the quoted results were obtained with F1 from a thermophilic bacterium
(40). Details of earlier work have been reviewed elsewhere (26, 67).
120◦ STEP PER ATP
For ATP concentrations, [ATP], from 20 nM to 6 mM, rotation
occurs in discrete 120◦ steps (2a, 64, 65). The time-averaged stepping rate depends
on [ATP] in a Michaelis-Menten manner, with a Vmax of 390 steps s−1 and Km of
15 µM (65). Statistical analysis of intervals between steps at various [ATP] indicates that each step is driven by the hydrolysis of one ATP molecule (2a, 64, 65).
The average stepping rate of single molecules above, however, is somewhat higher
than the hydrolysis rate measured in solution (65), apparently suggesting that the
coupling ratio, the number of 120◦ steps per ATP molecules consumed, is greater
than 1. Presumably, the hydrolysis rate is underestimated, because part of F1 in
solution is inhibited by MgADP (10, 18, 23, 26, 32), whereas a single-molecule
observation focuses on active molecules in the field of view (26). Our expectation
is that the coupling ratio is close to but slightly less than 1.
ANGLE-INDEPENDENT TORQUE When a micrometer-sized actin filament is
attached to γ , F1 rotates slowly because of the hydrodynamic friction against
the moving actin filament. The frictional drag coefficient, ξ , is given (2, 20) by:
ξ = (4π/3)ηL 3 /[ln(L/r ) − 0.447]
2.
for an actin filament of length L and radius r (∼5 nm) rotating around one end,
where η is the viscosity of the medium and is ∼10−3 Nm−2 s at room temperature.
(This equation was mistyped in some of our earlier work.) If, instead of actin, a
spherical bead of radius a is attached off axis such that the distance between the
bead center and rotation axis is x, ξ is given by:
ξ = 8π ηa 3 + 6π ηax 2 .
3.
The torque N needed to rotate the filament against the friction is given by:
N = ωξ,
4.
where ω is the speed of rotation (in radians s−1). From the rotation speed during
each 120◦ step, the torque F1 produces against the friction is estimated to be ∼44
pN · nm; the torque appears independent of the rotary angle, because the rotation
speed is constant during a 120◦ step (25). A similar value of 40 pN · nm is obtained
from the rotation speed averaged over many revolutions, under the condition of
high enough [ATP] in which F1 bearing an actin filament rotates smoothly without apparent steps (64). Thus, the work F1 can do in a 120◦ step amounts to 80–
90 pN · nm or ∼20 kBT, calculated as 40–44 pN · nm × (2/3)π radian (=120◦ ).
This is almost equal to the physiological |1G| for ATP hydrolysis above, suggesting that the energy conversion efficiency, the average work done in a 120◦ step
divided by |1G|, can reach ∼100% (but see below). The torque of ∼44 pN · nm
may be somewhat underestimated, because the friction near a glass surface
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Figure 3 Substeps in F1 rotation at 20 µM ATP (upper panel) recorded with a
temporal resolution of 0.125 ms (65). With this resolution, substeps are not discerned
at 2 mM ATP (lower panel), implying that the ATP-waiting dwell is <0.125 ms and the
next 90◦ rotation is complete within 0.125 ms of ATP binding; it also implies that F1 is
completely reset and ready to bind a next ATP by the end of a 30◦ substep. Horizontal
solid lines are separated by 120◦ , and dotted lines are drawn 30◦ below.
is higher than the estimate in Equations 2 and 3 (20). A slightly higher value,
50 pN · nm, has been estimated from the curvature of an actin filament attached
to γ (45). Here, the torque of F1 is balanced by the elastic recoil of a bent actin,
which, unlike friction, is a conservative force.
80◦ –90◦ SUBSTEP BY ATP BINDING AND 40◦ –30◦ SUBSTEP BY PRODUCT RELEASE
A
40-nm gold bead attached to γ (Figure 1) does not impede rotation, at least as
far as the average stepping rate is concerned, even though the bead is approximately four times as large as F1. Figure 3 shows time courses of the bead rotation
recorded at 8000 frames s−1 (65). At 20 µM ATP, the 120◦ steps are resolved into
∼90◦ and ∼30◦ substeps, as seen in most parts of the record. The dwell before
a 90◦ substep is much longer, on average, at lower [ATP], and the average dwell
is inversely proportional to [ATP]. This implies that the 90◦ substep is triggered
by binding of ATP. Because the 90◦ rotation is complete within 0.125 ms of ATP
binding (Figure 3, lower panel), most of the 90◦ rotation seems to be driven directly
by ATP binding, and not by a subsequent process such as ATP hydrolysis.
The dwell time before a 30◦ substep is independent of [ATP]. Analysis indicates
that the dwell consists of at least two reactions, each taking ∼1 ms (65). Because a
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30◦ substep is immediately (<0.125 ms) followed by a 90◦ substep at 2 mM ATP
(Figure 3, lower panel), the 30◦ substep completely resets the motor to the initial
ATP-waiting state. The simplest scenario is that the 30◦ substep is driven by the
release of a hydrolysis product, ADP or Pi, whichever is released last.
When F1 falls into the MgADP-inhibited state and stops rotating, it stalls at
the ∼80◦ position (18). This is consistent with the idea that the 30◦ substep is
associated with ADP release, of which the failure leads to inhibition. Or, the 30◦
substep may be driven by Pi release, which induces a conformational change in F1
(31). When rotation is driven by ATP-γ -S, which is slowly hydrolyzed by F1, videorate substeps of ∼80◦ and ∼40◦ are observed (50), indicating that ATP hydrolysis
occurs before the 30◦ substep and supporting the contention that ATP binding
drives most (80◦ ) of the 90◦ substep. A fluorescent ATP analog (Cy3-ATP) also
induces video-rate substeps of ∼80◦ and ∼40◦ in a mutant F1, and binding/release
of Cy3-ATP has been imaged simultaneously with the substep rotation (37a). This
study, combined with the ATP-γ -S study above (50), indicates that one of the two
∼1 ms reactions preceding the 30◦ substep is the ATP hydrolysis in a site that
bound ATP one step ago. These more recent results suggest that 80◦ rather than
90◦ is closer to the actual substep size, the difference being an experimental error,
or that the 80◦ rotation is driven by ATP binding and the remaining 10◦ by another
process such as ATP hydrolysis.
Potential Energy for γ Rotation
The scheme that emerges from these studies for the potential energy of γ rotation is
shown in Figure 4. We distinguish at least three states (A)-(C), (A) being the ATPwaiting state and (A0 ) being the next ATP-waiting state. For the chemical species
involved in rotation, two basic schemes have been proposed: a bi-site scheme in
which the occupancy of catalytic sites alternates between 1 and 2 (Figure 4b), and
a tri-site scheme alternating between 2 and 3 (Figure 4c). Previously, we suggested
that the bi-site scheme would be the norm because earlier studies (35) indicated
bi-site operation at least at low [ATP] and because the rotation rate is characterized
by a single Km down to 20 nM ATP (65). Site occupancy probed by a tryptophan
residue engineered in the catalytic sites, however, is more than 2 at ≥10 µM ATP,
and steady-state hydrolysis activity correlates well with the fraction of molecules
with an occupancy number of ≥2 (42, 47, 59). Weber & Senior (59, 60) thus
conclude that three-site filling is a must for rotation. A crystal structure in which
three sites are occupied with a nucleotide (34) also supports a tri-site scheme.
Work in our lab is as yet inconclusive: Rotation driven by Cy3-ATP proceeds in
a tri-site mode (37a), whereas analysis of fluorescence resonance energy transfer
apparently suggests bi-site if crystal structures of mitochondrial F1 are undistorted
by crystal packing (63). The average stepping rate is proportional to [ATP] down
to 0.6 nM ATP (N. Sakaki, unpublished results), suggesting that, if it operates in
a tri-site mode, two catalytic sites bind a nucleotide extremely tightly, apparently
in contradiction to the binding data above. Boyer (11) suggests that bi-site is
fundamental, but retention of the ADP to be released may lead to three-site filling.
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Figure 4 Potential energy for γ rotation deduced from the substep behavior (65).
(a) Schematic diagram of 90◦ and 30◦ substeps. (b) Corresponding nucleotide states
in a bi-site scheme. The central arrow shows the orientation of γ . ATP∗ represents
ATP or ADP + Pi; ADP may be ADP + Pi or Pi. (c) A tri-site scheme. Light shade
indicates difference in site occupancy. (c0 ) A tri-site scheme that is closer to b is also
possible, in which site occupancy remains 2 and becomes 3 only during the 90◦ substep.
(d) Potential energies for the four chemical states (A)-(A0 ). The γ orientation in state
(A) is taken as 0◦ . The vertical shifts of the potential curves in this figure are arbitrary.
See Figures 7 and 8 and associated explanations in text.
ATP binding may indeed assist release of a retained ADP (K. Adachi, unpublished
data) (Figure 4c0 ). More work is needed before concluding firmly that three sites
must be filled for rotation to occur. In this article, we define states (A) and (A0 )
in Figure 4 as the ATP-waiting state, and (B) as the state immediately after ATP
binding, regardless of the actual occupancy number.
The scheme in Figure 4a and the angle-independent torque of 40–44 pN · nm
suggest a potential energy diagram for γ rotation in each chemical state, as in
Figure 4d. Here, solid lines reflect the experimental constant torque, and dashed
lines are added to provide minima at the experimentally observed stationary angles.
In the ATP-waiting state (A), γ rests at the potential minimum at 0◦ . Binding of
ATP changes the state to (B), whereupon γ slides down the constant slope toward
the minimum at 90◦ . The chemical transition from (B) to (C) is mechanically
almost silent. Release of the last hydrolysis product restores state (A0 ), whereupon
γ slides down the slope for the last 30◦ . As noted above, the minima for (B) and
(C) may well be closer to 80◦ , or the minimum for (B) near 80◦ and that for (C)
near 90◦ .
F1 rotation can be driven by GTP and ITP, but not by CTP and UTP at concentrations up to 300 µM (38). The Michaelis-Menten constant, Km, for hydrolysis
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differs among ATP, GTP, and ITP, but all these purine nucleotides drive an actin
filament at similar speeds, implying similar torque. Thus, it appears that the
potential energies in Figure 4d are the properties intrinsic to F1, and nucleotides
are either able or unable to drive transitions among the potential curves.
How an ATP-Driven Molecular Machine May Work
The potential diagram in Figure 4d is our experimental answer to the prime question
of how a molecular machine can convert free energy of ATP hydrolysis into work.
We believe that protein machines work by changing their conformations. In the
case of F1, rotation of γ is in itself a sequence of conformational changes of F1,
and individual β subunits, in particular, undergo large conformational changes in
response to nucleotide binding (Figure 2). Each potential curve in Figure 4d thus
represents the change in conformational energy of the protein (F1) in that chemical
state, as γ is rotated to designated angles. Before discussing how these potentials
could explain rotation, let us summarize in Figure 5 what the experiments have
told us. A protein molecule, shown in green, is to do mechanical work by changing
its conformation between the upright posture and bent one.
MAJOR WORK IS DONE BY ATP BINDING, AND SOME BY PRODUCT RELEASE
At
least in F1, the major, work-producing conformational change (80◦ –90◦ rotation)
is driven by ATP binding. Another change, back to the original conformation upon
product release, also does some positive work. ATP hydrolysis per se does not
contribute much to work output. In other words, major free-energy drops, which
can be converted to work, occur upon ATP binding and product release (brown
arrows in Figure 5). Indeed, on a protein machine such as F1 and myosin, ATP and
its hydrolysis products ADP + Pi are near equilibrium, as demonstrated by spontaneous ATP synthesis on these proteins in the presence of high concentrations of
ADP and Pi (30, 48, 61, 66). Exploiting binding and release, rather than hydrolysis, seems natural if one is to design an efficient protein machine: Upon substrate
binding, many bonds are newly formed between the substrate and protein, whereas
hydrolysis would result mainly in bond reorganization without a gross change in
the total number of bonds.
NUCLEOTIDE AFFINITY CHANGES AS ROTATION (CONFORMATIONAL CHANGE) PROCEEDS If ATP binding drives a conformational change in a protein molecule,
that particular conformational change will accompany an increase in the affinity
for ATP, by the law of action and reaction. Likewise, if ADP (Pi) release drives
another conformational change, that will accompany a decrease in the affinity for
ADP (Pi). These points are crucial in understanding the mechanism of the F1 motor
and molecular machines in general.
A highly schematic explanation is given in Figure 5 for the case of ATP binding.
Starting from the upper left, initial binding of ATP (light color) is accomplished
by the formation of multiple weak bonds (violet dots). The fact that ATP binding
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induces bending (conformational change) of the protein implies that the ATP
molecule pulls the protein toward it by forming more bonds (second from left).
As a result, the ATP molecule in the bent protein is now tightly held in the bent
protein, implying higher affinity. Constant torque can be explained by sequential
increase in the number of bonds. A more realistic view for the particular case of
F1 has been given by molecular dynamics simulation (3).
The magnitude of affinity changes can be estimated by referring to Figure 4d
(65). As γ is rotated from 0◦ to 90◦ , curve (B) for the ATP-bound state becomes
lower (more stable) than curve (A) for the state before ATP binding. The stabilization upon the 90◦ rotation amounts to >60 pN · nm in energy, or to an increase in
the affinity for ATP of more than exp(60 pN · nm/kBT) ∼ 2 × 106-fold. Similarly,
rotation from 90◦ to 120◦ is accompanied by a decrease in the affinity for ADP
(Pi) of more than exp(20 pN · nm/kBT) ∼100-fold.
THE PRIMARY ROLE OF HYDROLYSIS IS TO RESET THE MACHINE If work is done
by binding and release, why is ATP hydrolyzed? The reason is that, after the
ATP-driven conformational change, the ATP molecule is so tightly bound that the
reaction stops there (second from left in Figure 5). To remove this blocking ATP,
the protein machine splits it into two entities so that individual products, each held
by a smaller number of bonds, can escape one by one.
In the case of F1, which synthesizes ATP efficiently upon reverse rotation, we
propose (25, 26) that hydrolysis may well produce some work (rotation), although
direct evidence is not yet available. The model below considers this possibility.
A CAMSHAFT MODEL OF F1 ROTATION
Here we present a simple mechanical model (Figure 6) that is based on the models of Oosawa & Hayashi (44) and of Wang & Oster (57). The purpose is to
show a relatively concrete example of the mechanism of chemo-mechanical energy conversion that mimics some of the essential features of the F1 motor. For the
purpose of illustration, we design the model to work in a bi-site scheme at moderate ATP concentrations, because a smaller number of chemical pathways need
to be considered in bi-site. Most of the discussion below, however, is independent
of the details of chemical pathways and thus is applicable to a tri-site scheme as
well.
A Push-Pull Mechanism for Torque Generation
As seen in Figure 2, crystal structures of mitochondrial F1 (1, 15) have shown
that a β subunit binding a nucleotide is bent in such a way that its upper portion
pushes γ toward and beyond the central axis. An empty β, on the other hand,
retracts and pulls γ toward it; an isolated β subunit (62) or βs in an α 3β 3 complex without γ (51) adopt a conformation similar to the empty β in the whole F1,
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although a recent simulation shows otherwise (8). Wang & Oster (57) have proposed that these push-pull actions against the slightly slant and skewed γ constitute
the driving force of rotation. This is consistent with the fact that, in the upper and
lower portions of the central cavity of the α 3β 3 cylinder where γ makes extensive
contacts with the cylinder wall (Figure 2, top), the cylinder wall is covered with
hydrophobic residues (1), allowing lateral slippage while responding to push and
pull. The contact at the γ tip (at the lower end of and as illustrated in the top part of
Figure 2) seems relatively unimportant, because up to 12 residues can be removed
without affecting torque (36). Another major contact is made between γ and the
top surface of a β, near the orifice. The contact area of β, the so-called DELSEED
motif (yellow atoms in Figure 2, top, left), contains a cluster of conserved acidic
residues, suggesting a role in torque generation. Complete elimination of the negative charges, however, did not affect the rotational characteristics (16), although
molecular dynamics simulation suggests a role for this region (29). Thus, we also
assume that the push-pull actions by β are the major source of the torque, and
we model γ as a camshaft and let bending/unbending of β push/pull the camshaft
(Figure 6a).
The push, or bending of β, is driven by ATP binding (Figure 6b), and conversely
a bent β shows a higher affinity for ATP than an unbent β does. The pull, or
unbending of β, is driven by the release of ADP (Figure 6c), which is assumed
to dissociate last, and unbending leads to a lower affinity for ADP (and Pi). The
central camshaft is so designed that ATP-driven bending of a β rotates the γ shaft
by 80◦ (Figure 6e, e1–e10), and unbending of another β upon ADP release drives
the last 30◦ from 90◦ to 120◦ (Figure 6e, e13–e16). By repeating the sequence in
Figure 6e, the central γ shaft continues to make counterclockwise 120◦ steps (see
Supplementary Movie 1; follow the Supplemental Material link from the Annual
Reviews home page at http://www.annualreviews.org).
The Mechanism Warranting Unidirectional Rotation
In principle, simple pushes and pulls would rotate the shaft in either direction. In
our model, there are two stages in which the γ shaft could rotate clockwise (wrong
direction). The first is the situation in Figure 6e1, in which two βs are empty:
Binding of ATP to the wrong β (β 3) would lead to clockwise rotation. Binding
to the wrong β, however, is prevented by virtue of the slope of the camshaft. The
radii of the cam are varied in such a way that the correct β (β 2) is already bent
slightly, whereas β 3 is completely unbent. Thus, the correct β has a higher affinity
for ATP, and thus, in most cases, this β binds the next ATP. In a tri-site scheme,
the situation of two βs being empty does not arise. However, when ADP is to be
released, one of three filled sites needs to be selected. This is equivalent to the
choice of an ATP binding site, because ATP will bind to the site from which ADP
has dissociated.
The next critical stage is in Figure 6e10, in which two βs bind ATP tightly and
either ATP could be hydrolyzed. If the β that has just bound ATP (β 2) hydrolyzes
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the ATP, the shaft would make a backward substep upon release of the hydrolysis
products. To prevent this uncoupling scenario, we postulate (as in References 25,
26, and 44) that hydrolysis of ATP drives a small amount of counterclockwise
rotation of γ (Figure 6d). That is, the tip of the toy β is twisted to the right
(as it faces γ ) when its bound ATP is split into ADP + Pi, and further as Pi is
released. The law of action and reaction states, then, that twisting the tip of β to the
right drives the equilibrium between ATP and ADP + Pi toward hydrolysis, and
twisting to the left shifts the equilibrium toward ATP synthesis. Once we assume
the coupling between the tip motion and hydrolysis/synthesis reactions, rotation
in the correct direction is warranted: In Figure 6e10, β 1 can hydrolyze ATP by
twisting its tip to the right, but β 2 cannot because the tip motion is obstructed by
the cam wall. A theoretical calculation has indicated that a β that presumably has
just bound ATP favors ATP over ADP + Pi (13).
Clockwise Rotation Leads to ATP Synthesis
The model predicts synthesis of ATP when the γ shaft is forcibly rotated clockwise. This can be seen by following Figure 6e in the reverse order (also see
Supplementary Movie 2; follow the Supplemental Material link from the Annual
Reviews home page at http://www.annualreviews.org). Clockwise rotation pulls
in and thus bends β 1, thereby increasing its affinity for ADP (Figure 6e, e16–e14).
Here, bending β 1 also tends to increase its affinity for ATP, but because the slope
of the camshaft twists the tip of β 1 to the right, the affinity for ADP is higher. Thus,
ADP and then Pi bind (Figure 6e, e13–e12). Further rotation twists the β 1 tip to the
left (Figure 6e, e11–e10) to facilitate the conversion of ADP + Pi to ATP, which
remains tightly bound. At this angle, the ATP that has been synthesized in the
previous step is still tightly bound by β 2, but further clockwise rotation (Figure 6e,
e10–e1) pries the catalytic site open and releases the ATP. Clockwise rotation of
γ in a molecular dynamics simulation has demonstrated that the affinity for ATP
decreases with rotation, as expected (7).
Recently, we have shown that ATP is indeed synthesized when the γ shaft of
actual F1 is forcibly rotated clockwise by electric magnets, through a magnetic
bead attached to γ (22). An external torque applied to one point on the motor
suffices to reverse the chemo-mechanical linkage and force the chemical reaction
in remote catalytic sites to proceed in reverse, increasing chemical free energy.
This is not a trivial result: consider whether one can pull back a linear motor, such
as myosin or kinesin, to let it synthesize ATP, and, if one can, where to pull.
POTENTIAL DIAGRAMS FOR THE CAMSHAFT MODEL
Here we derive potential diagrams for the camshaft model, which basically reproduce the experimental potential diagram in Figure 4d, to discuss the motor’s
behaviors under various 1G and under various external loads.
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Free-Energy Diagrams
The three sets of curves at the bottom of Figure 7a show the angle-dependent
energy, ψ Li (θ), that we assume for the interaction between γ and a β binding a
ligand, L, where L is ATP (T), ADP + Pi (H), ADP (D), Pi (P), or none (E); and
i = 1, 2, or 3 designates one of the three βs. These potential energies reflect the
shape of the camshaft. We assume that an empty β is stable when it is unbent and
that a β binding a ligand is stable when it is bent. Thus, ψ T1 (θ ), for example, is low
for θ between −30◦ and 80◦ , where β 1 remains fully bent; and ψ T1 (θ ) is highest
at θ = 100◦ , where β 1 with its tip twisted to the left is forced to unbend fully.
ψ Li (θ) for L = H, D, and P have the same shape as ψ Ti (θ ), except that their heights
are lower and they are shifted to the right, reflecting the twist of the β tip. ψ Ei (θ )
L
is assumed to be a mirror image of ψ D
i (θ ) except for a lower height. ψ i (θ ) for
◦
different i are related by translation of ±120 .
(θ ) for γ rotation is assumed to be given
The overall potential energy Ψ L1L2L3
0
by the simple sum of three energies:
(θ) = ψ1L1 (θ ) + ψ2L2 (θ ) + ψ3L3 (θ ).
Ψ L1L2L3
0
5.
Experiments with a chimera of fast and slow βs indicate that, at least for kinetics,
the properties of individual βs are additive (4). As seen in Figure 7a (top), the total
potential energies in Equation 5 essentially reproduce the experimental potential
diagram in Figure 4d: Ψ TEE
0 (θ), for example, represents the ATP-waiting state, and
Ψ TTE
0 (θ) represents the state after ATP binding. In fact, the shape and height of
the individual potential, ψ Li (θ), have been chosen arbitrarily, solely to reproduce
the experimental diagram. The choice is nothing but one out of many possibilities
that could underlie the experimental potentials.
In Figure 7a (top) and in all subsequent potential diagrams including Figure 7b,
a constant is added to each potential energy such that each curve represents the free
energy of the system including the motor and its environment. That is, starting from
an arbitrarily chosen initial state, the following constants are added (subtracted for
a reverse reaction) for every chemical transition in a β.
ATP
− kB T ln[ATP]
E → T : −lnK a0
6a.
T → H: −lnK hyd0
6b.
Pi(1st)
+ kB T ln[Pi]
H → D: −lnK d0
6c.
H → P:
ADP(1st)
−lnK d0
+ kB T ln[ADP]
6d.
ADP(2nd)
+ kB T ln[ADP]
D → E: −lnK d0
6e.
Pi(2nd)
+ kB T ln[Pi],
P → E: −lnK d0
6f.
where K0 is the equilibrium constant for the forward reactions (in the absence of
the interaction between β and γ ). These constants satisfy the conditions:
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Pi(1st)
ADP(2nd)
ADP(1st)
Pi(2nd)
ATP
ATP
K a0
· K hyd0 · K d0
· K d0
= K a0
· K hyd0 · K d0
· K d0
= exp(−1G 0 /kB T ),
7.
where 1G0 is the standard free-energy change for ATP hydrolysis.
For example, starting from TEE, angle-dependent free energies for subsequent
states are given by:
Ψ TEE (θ) = Ψ TEE
0 (θ)
8a.
ATP
Ψ TTE (θ) = Ψ TTE
0 (θ) − lnK a0 − kB T ln[ATP]
8b.
ATP
(θ) − lnK a0
− kB T ln[ATP] − lnK hyd0
Ψ HTE (θ) = Ψ HTE
0
ATP
(θ) − lnK a0
− kB T ln[ATP] − lnK hyd0 −
Ψ DTE (θ) = Ψ DTE
0
8c.
Pi(1st)
lnK d0
+ kB T ln[Pi]
8d.
Pi(1st)
ATP
Ψ ETE (θ) = Ψ ETE
0 (θ) − lnK a0 − kB T ln[ATP] − lnK hyd0 − lnK d0
ADP(2nd)
+ kB T ln[Pi] − lnK d0
+ kB T ln[ADP]
8e.
= Ψ ETE
0 (θ) + 1G 0 + kB T ln[ADP][Pi]/[ATP]
8f.
= Ψ ETE
0 (θ) + 1G.
8g.
Note that Ψ ETE(θ) for ETE obtained by unbinding of ATP from β 1 and binding
TEE
◦
of ATP to β 2 is given simply by Ψ ETE
0 (θ ), which is Ψ 0 (θ − 120 ), because
hydrolysis is not involved.
For the equilibrium constants, we choose arbitrarily the following values:
= 6.1 × 1011 M−1, Khyd0 = 0.13, KPi(1st)
= 2.0 × 10−3 M, KADP(2nd)
=
KATP
a0
d0
d0
Pi(2nd)
−5
1.5 × 10−3 M, KADP(1st)
=
1.0
×
10
M,
and
K
=
0.31
M.
Note
that
these
d0
d0
are just parameters and that actual equilibrium constants exhibit angle dependence
determined by the difference between the initial and final ψ(θ )s involved in the
relevant chemical transition. For example, the affinity KATP
a1 (θ ) of β 1 for ATP is
given by:
©£
¤
ª
ATP
ATP
(θ) = K a0
· exp ψ1E (θ ) − ψ1T (θ ) /kB T ,
9.
K a1
which takes a maximal value of 6.7 × 1014 M−1 at θ between −10◦ and 80◦ and
a minimal value of 2.6 M−1 at θ = 120◦ . The angle dependence can readily
be appreciated from the vertical difference between the relevant two curves in
Figure 7a (top). For example, Khyd(θ ), which is related to the difference between
HTE and TTE curves, remains low and constant (=0.13) between −20◦ and 80◦ ,
and takes high values favoring hydrolysis between 80◦ and 120◦ , reaching a maximal value of 7.0 × 105 at 100◦ . Note that the angle-dependent equilibrium constants
also satisfy Equation 7 at each angle. For example, to bind ADP (not ATP) and
ADP
, and KPi
Pi from the medium for ATP synthesis, KATP
a , Kd
d need all be small at
some angle, which requires that Khyd be large at the same angle. In our model, this
happens around 90◦ for β 1.
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Major Reaction Pathways
Now we discuss the rotation mechanism on the basis of the free-energy diagrams.
In Figure 7a (top) we show the major reaction pathway at the indicated nucleotide
concentrations. We choose TEE to be the starting state, where the potential minimum is at 0◦ and the motor waits for ATP at this angle. When ATP binds to the
correct site (on β 2), the potential shifts to the red TTE curve. At 0◦ , however, the
TTE curve is above the TEE curve, indicating that ATP binding is an uphill reaction.
This is because the chosen ATP concentration, 2 µM, is low; the motor has to wait
long for ATP to bind. The uphill part only poses a kinetic barrier, because as rotation
proceeds the TTE curve becomes lower than TEE, stabilizing bound ATP. Indeed,
Equation 8b indicates that the rate of passage over the (entropic) barrier at 0◦ , or
the rate of ATP binding, should be proportional to [ATP], which is the expected
kinetics. It is possible, however, that the motor binds ATP by crossing from TEE
to TTE around 15◦ , where the barrier height is lower. This could happen through
thermal fluctuation of the γ angle, and the rate of ATP binding would no longer
be proportional to [ATP]; kinetic barrier in this pathway is actually high, because
ATP must arrive at the moment where γ happens to fluctuate counterclockwise by
∼15◦ . Experimentally, the stepping rate of the F1 motor is proportional to [ATP]
down to <1 nM, suggesting that ATP binding occurs by an upward jump from
TEE to TTE. The rate of binding of a fluorescent ATP analog indeed becomes
lower when γ is rotated slightly counterclockwise from an ATP-waiting angle
(K. Adachi, unpublished data). It seems that closure of the binding site poses a
kinetic barrier and impedes binding, in addition to stabilizing a bound nucleotide;
higher affinity for ATP upon counterclockwise rotation results from decrease in
the rate of ATP release.
Once ATP is bound, the motor slides down the slope of the TTE curve to
80◦ , producing an apparent torque of ∼50 pN · nm. That is, an actin filament or
bead attached to γ can be rotated at a speed commensurate with this much torque
(Equation 4). Note that in the world of molecular machines, mass is negligible,
or motion is always overdamped, and thus a linear potential produces motion at a
constant speed. When γ reaches 80◦ , the motor again waits for thermal activation
to hydrolyze ATP and, upon activation, switches onto the HTE curve to reach
90◦ . One of the two experimentally observed ∼1 ms reactions around 90◦ likely
corresponds to this process. At 90◦ , both PTE and DTE curves are below the HTE
curve, because [ADP] and [Pi] are low in this diagram, and thus either ADP or Pi is
readily released. The ETE curve is further below, and the motor lands on this curve
by releasing the last hydrolysis product. The last 30◦ is again a torque-generating
downhill process, except for the first 10◦ , which is slightly uphill and which might
correspond to the observed second ∼1 ms reaction. At 120◦ , the motor waits for
the next ATP.
In our scheme, torque is given as the slope of the potential energy, which is an intrinsic property of the motor. Thus, the torque is independent of speed or nucleotide
concentrations. A cargo attached to the rotor rotates at a speed commensurate
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with Equation 4, as long as internal impeding factors such as friction are negligible. Indeed, the apparent torque calculated from Equation 4 is independent
of [ATP] and of speed up to ∼30 revolutions s−1, compared with the full speed
of 130 revolutions s−1 (64, 65): Internal friction does not manifest its effect up
to one fourth of the maximal speed. Another rotary motor, the flagellar motor
that is driven by proton flow, also shows nearly speed-independent torque at low
speeds (5).
Other Pathways
We consider five chemical states, T, H, D, P, and E, for each β, and thus there
are a total of 125 chemical states, each associated with a unique potential energy,
Ψ 0(θ). More states would appear in our free-energy diagram showing Ψ (θ)s, in
which the same chemical state can have a different vertical offset, in multiples of
1G, depending on the net number of ATP hydrolyzed/synthesized to reach that
state. In Figure 7a (top) we present only five of these Ψ (θ)s. Now we show that
these indeed constitute the major pathway, at least at moderate concentrations of
nucleotides.
In Figure 7b we show all states that are accessible from TEE (bottom) or TTE
(top): Destinations from T are E (unbinding) or H (hydrolysis), and destinations
from E are T, D, or P (all bindings). From TEE at 0◦ (an ATP-waiting state), TTE at
80◦ is the most likely destination because the barrier to TTE is low (almost absent
at the indicated ATP concentration) and the minimum of TTE at 80◦ is the lowest.
This is why we show the path TEE → TTE as the major pathway in Figure 7a
(top). The other, equally stable destination is TET at −40◦ (binding to the wrong
site), but a high barrier exists because β 3 is fully unbent and its affinity for ATP is
low. Binding of ATP to this wrong site could still happen rarely and might account
for occasional backward 120◦ steps observed at low ATP concentrations (64). In
the model, however, the major path from TET at −40◦ is to return to TEE at 0◦ ,
without hydrolysis of ATP. Completion of a backward 120◦ step through hydrolysis
of ATP, i.e., reaching EET at −120◦ through HET and DET, is a minor path.
Another path from TEE at 0◦ is toward HEE at 0◦ (hydrolysis without rotation).
Because the barrier to HEE is only a few kBT high, the motor fluctuates between
the two states, or ATP and ADP + Pi are near equilibrium at 0◦ . This can account
for the extensive exchange of Pi oxygens during catalysis at low [ATP] (10). From
HEE, there is another pathway to EEE through DEE or PEE. This is the so-called
uni-site catalysis (26) and does not accompany rotation in our model, in accord
with the experiment showing ATP hydrolysis in F1 in which γ is cross-linked to
β (14).
Thus, although the pathway shown in Figure 7a (top) is the major one in that it
crosses the lowest barriers and follows free-energy minima, there are many other
paths that can be populated to some extent. Some are just detours that return to
the major pathway without expenditure of ATP, as in the example of TEE →
TET → TEE or TEE → HEE → TEE. Others constitute uncoupled pathways in
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which ATP hydrolysis occurs without rotation, such as the uni-site hydrolysis or
reverse rotation failing to synthesize ATP (see below). It is also possible that a
forward 120◦ step is made without expending ATP, as in the example of TEE to
ETE simply by binding and unbinding of ATP. A backward 120◦ step, however,
occurs through an equivalent sequence toward EET, with the same (and small)
probability: On the free-energy diagram the three states, TEE, ETE, and EET, are
related by a purely horizontal 120◦ shift unless ATP hydrolysis or synthesis is
involved. Naturally, there is no unidirectional motion without expenditure of free
energy. The extent to which the nonmajor pathways are populated depends on the
rate constants of chemical reactions and rotation. We do not discuss kinetics here
because too many angle-dependent rates are involved and most are experimentally
unknown. Formalisms for kinetic analysis are given in References 44 and 57.
EFFICIENCY OF CHEMO-MECHANICAL ENERGY
CONVERSION
Hereafter, we focus on the major pathway shown in Figure 7a (top), and we further
neglect PTE to simplify the diagram. The discussion below is based on free-energy
diagrams that are basically experimental, and does not depend on the details of the
model, such as whether it operates in the bi-site or tri-site scheme. TEE implies
an ATP-waiting state, TTE the state after ATP binding (to an unspecified site),
HTE after hydrolysis of ATP (in an unspecified site), DTE after Pi release, and
ETE the ATP-waiting state after a 120◦ step accompanying hydrolysis of one ATP
molecule has occurred. An alternative representation for a tri-site pathway, among
others, is TED → TTD → TTE → HTE → DTE, which is similar to the bi-site
scheme except that the last 30(40)◦ substep is driven by Pi release and that ADP
stays on F1 until prompted by subsequent ATP binding (Figure 4c0 ). Another, more
genuine tri-site scheme is TEH → TTH → HTH → HTD → HTE, in which ATP
binds to β 2, hydrolysis occurs in β 1, and product release is from β 3.
Responses to Decreasing 1G in the Absence of External Load
Figure 8a shows a free-energy diagram for nucleotide concentrations at which the
free energy decreases relatively smoothly and linearly as rotation proceeds. When
a large cargo such as an actin filament or bead is attached to γ , the motor will do
an apparent work of ∼100 pN · nm or 25 kBT per 120◦ step against the viscous
load, which is approximately equal to the free-energy input, 1G, under the chosen
conditions. The situation, however, is different from work against a conservative
force (see below), and the cargo simply stirs the solution, producing heat. Net heat
production per ATP hydrolyzed, including the heat associated with the chemical
reactions and possible loans from the environment, is not 100 pN · nm and is given
by −1H, which is ∼45 pN · nm at pH 7 (28). The heat output is independent of
1G or of whether a cargo is attached to γ , unless the motor does work against a
conservative force.
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Now we reduce 1G, first by decreasing [ATP] (Figure 8b). In the free-energy
diagram, all curves after ATP binding are brought up by the term −kBT ln[ATP]
(Equations 8b–e). ATP binding is now uphill, and the motor must borrow some
free energy (negative entropy) from the environment. That is, the motor must wait
at ∼0◦ for the rare event of ATP binding.
We can further reduce 1G by increasing [ADP] and [Pi] (Figure 8c). Then the
curve DTE is brought up by the term kBT ln[Pi], and ETE is further brought up
by the term kBT ln[ADP]. The result is that both Pi and ADP releases are uphill:
Immediate rebinding occurs in proportion to [Pi] and [ADP], which are both high.
Thus, the motor cannot readily release the products and would dwell around 90◦ .
However, in the rare event that the motor happens to release both products, rotate
to ∼120◦ , and bind a next ATP, it can proceed onto the curve ETT and reach 200◦
(dashed curve in Figure 8c). Although the probability of the forward stepping
should be extremely low under the chosen conditions, it is still much higher than
the reaction back to TEE at 0◦ . As long as 1G is negative, the motor eventually
rotates counterclockwise however rarely it may step.
Changes in the rates of ATP binding and product rebinding enable the motor
to “sense” the nucleotide concentrations, or free energy of the environment. Note
that, irrespective of 1G, rotation occurs along the same slopes and thus at the
same speed. If one calculated the amount of apparent work from the rotary speed
during forward stepping, it would still be ∼100 pN · nm, whereas 1G is only
about −20 pN · nm in Figure 8c. The apparent efficiency could thus exceed 100%.
Thermodynamic efficiency, however, has to be discussed for the situation when a
motor works against a conservative force that pushes back the motor (see below).
To deal with the case of viscous load, the concept of generalized efficiency has
been introduced (69) and developed as Stokes efficiency (58), which is based on
the rotary speed averaged over a long time including interstep dwells and which
is bound by 100%; for a high Stokes efficiency, the motor torque must be angleindependent, as in the F1 motor. Experiments at varying [ATP] have shown that the
apparent amount of work in a 120◦ step is independent of [ATP], as discussed above
(64). Experiments at high [ADP] are difficult because of the MgADP inhibition.
Rotation Against a Conservative Force
A different scenario arises when an external torque is applied to the motor. A
conservative torque, such as one exerted by a magnet or bent actin, operates on a
motor regardless of whether the motor is moving. The magnitude of torque depends
on the motor angle, not speed. In contrast, the viscous torque disappears when the
motor stops. The viscous torque impedes motor movement, but it never pushes
back the motor, whereas a conservative torque does.
Below we examine the case of angle-independent external torque (Figure 8d–i).
The torque Fo applies on F1 is expected to be relatively smooth, because the rotor
of Fo possesses 10- to 14-fold symmetry (13a, 53, 67). An externally applied force
on a linear molecular motor, e.g., with an optical trap, is practically constant over
the size of the motor step.
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Figure 8d–f shows diagrams of free energy (including the potential energy of the
external torque) at increasing opposing torque for the nucleotide concentrations
in Figure 8a. As seen, the motor moves forward (counterclockwise) as long as
the total free energy decreases over a 120◦ step (TEE → ETE), or as long as
1G + W is negative, where W (>0) is the work done against the external opposing
torque Next (W = Next · 2π/3). That is, the motor can do positive work (output) up
to the amount, −1G (input), determined by the nucleotide concentrations. When
W exceeds −1G (Figure 8f), the motor rotates in reverse, pushed by the external
torque. The reverse rotation leads to ATP synthesis, basically in the sequence
ETE → DTE → HTE → TTE → TEE, although under the conditions of Figure
8f, the motor tends to hop between these states. A more comfortable situation for
synthesis is shown in Figure 8g, in which the above sequence is almost guaranteed
(thanks to the higher concentrations of ADP and Pi).
Here we discuss the efficiency of chemo-mechanical energy conversion, first
considering solely the major pathway; the discussion ignoring uncoupled pathways
sets an upper limit for the efficiency. For forward rotation, the input is −1G,
which is determined by nucleotide concentrations, and the output is W, which
is determined by the external torque. The efficiency of chemical to mechanical
energy conversion would thus apparently be W/|1G|, which is independent of the
motor properties except for the step size. The efficiency would approach 100%
as W and |1G| become comparable. When W equals |1G|, the motor fluctuates
equally in both directions, hydrolyzing and resynthesizing ATP. When W > |1G|,
the efficiency of mechanical to chemical energy conversion (ATP synthesis) would
be given by |1G|/W, again independent of motor properties. When W < |1G|,
the difference is dissipated as an increase in entropy (and heat if W < |1H| =
45 pN · nm). When W > |1H|, the difference in energy is supplied, in the form of
heat, by the environment as the heat bath.
According to the argument above, a motor would always achieve 100% efficiency when W equals |1G|. In fact, there are many uncoupled pathways that
reduce the efficiency and that critically depend on the make of a particular motor.
The efficiency above, W/|1G| or |1G|/W, is thus an upper limit. Actual efficiency depends on kinetic parameters (rate constants) that dictate the probabilities
of branching among various pathways.
Uncoupling scenarios can be seen even on a diagram depicting the major pathway. In Figure 8f, for example, there is a possibility that the motor, starting on the
ETE curve at 120◦ , slides along the ETE line toward the left without picking up
ADP and Pi. It encounters a small barrier at 0◦ , but there is a certain probability
that the motor crosses the barrier toward left. That is, an external torque may simply rotate the motor clockwise without synthesizing ATP. The probability of this
scenario is high under a higher torque, as one expects: If one rotates the motor too
fast, the motor will certainly fail to pick up ADP and Pi or to synthesize ATP.
Another example is given in Figure 8h, in which |1G| is quite large (136
pN · nm) and might be expected to drive counterclockwise rotation against the large
external torque of 60 pN · nm (W = 123 pN · nm). The shape of the TEE (or ETE)
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curve, however, is such that the motor yields to the external torque and tends to
rotate clockwise without hydrolyzing or synthesizing ATP. There must be an upper
limit to the force (torque) a motor can produce, and it is about 50 pN · nm for this
particular motor. Near the upper limit, the tendency toward uncoupled, backward
motion increases, as seen in Figure 8e. One cannot let a motor do an arbitrarily
large amount of work by increasing |1G|. “Give” (yielding under a high backward
force) is a common phenomenon for linear molecular motors in an optical trap (33,
54).
Dissipation and Friction
Let us confine the discussion to the case of coupling ratio being close to 1 (see
Reference 46 for more general discussion). Then, the efficiency is close to W/|1G|
or |1G|/W, determined by |1G| and W alone. This implies that the magnitude of
friction, internal or external, has little effect on the efficiency. A poorly designed
motor with high internal friction simply runs slowly, with the same efficiency as
a low friction motor. The efficiency also remains the same if an external friction
is added, by attaching a large cargo to the rotor. The cargo only slows down the
motor.
One might think that dissipation could occur via free-energy drop without motor
motion, such as the initial ATP binding, before subsequent rotation, at high [ATP].
In Figure 9a, for example, the overall 1G is assumed to be the same as in Figure
ADP
8a, but [ATP] (or KATP
a0 ) is set high and [ADP] (or Kd0 ) is set correspondingly
low. Thus, ATP binding (downward arrow) dissipates free energy because no work
is done during the binding, except for the internal work on the protein machine
that is independent of [ATP]. The dissipation, however, is canceled at the moment
of ADP binding (upward arrow), which is now uphill and requires a loan of free
energy from the environment. Again, the overall efficiency is unchanged, although
the motor runs slower because of the uphill portion.
HOW TO DESIGN AN ATP-DRIVEN MOLECULAR MACHINE
In principle, one could conceive of a motor that makes multiple steps per ATP
hydrolyzed (27); the motor could even be smart enough to make only one step
under a high load to produce a correspondingly high force. Though attractive,
working out a possible, reasonably detailed design for such a motor is not easy for
us. We therefore restrict our discussion to one-step motors.
We presume that the most desirable characteristics of a molecular machine are
high efficiency of energy conversion and high speed. If so, the design shown in
Figure 8 is close to ideal, as an ATP-driven rotary motor under the nucleotide
concentrations of Figure 8a, d–f and as a torque-driven ATP synthesizer under the
nucleotide concentrations of Figure 8g.
Linear potentials that produce angle-independent torque, or position-independent force for a linear motor, are better than curved potentials. Figure 9b, for
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example, shows parabolic potentials (solid lines), which may be expected for a
linear molecular motor (such as kinesin) that makes a diffusional search (shallow
potential on the left) followed by binding to a specific site (deep potential on the
right). Such a design, however, suffers from extreme slowing down in the presence
of an opposing load, because the barrier between the two potentials rises sharply
(dotted lines). Linear potentials do not suffer from this effect (Figure 8d–f). Linear
potentials are also advantageous when a molecular machine is to be driven by an
external force (Figure 8f), because they allow smooth, less bumpy and thus faster
motion.
To obtain a high efficiency of energy conversion at a high speed, free energy must
decrease smoothly, as in Figure 8a, without an uphill portion that degrades speed
and without a vertical drop that degrades the efficiency. Even with a well-designed
motor, however, these conditions are met only under one specific set of nucleotide
concentrations. In other words, a molecular machine must be designed to work
best in its working environment; its performance drops in other environments.
If one is to avoid uncoupling, or give, at a high load, one needs a barrier(s)
against uncoupled pathways. This problem is serious for a machine to be driven
by an external force, such as the ATP synthase. Potential diagrams that are too
smooth, such as those in Figure 8f or 8h, are liable to uncoupling. Ideally, barriers
should be introduced in positions outside the normal pathway, such as the small
bumps around −20◦ in Figure 8f. An arbitrarily high barrier, however, cannot be
introduced, because it must disappear in another chemical state (to let the motor
cross that point); the appearance/disappearance of the barrier must be driven by a
chemical reaction with a limited drop in free energy. A solution that sacrifices speed
is to allow the presence of some bumps or uphill portions in the major pathway, as
in Figure 8g. A highly bumpy diagram, as shown in Figure 8i, warrants tight coupling (and ∼100% energy-conversion efficiency near the balancing |1G|, which is
30 pN · nm in Figure 8i), although speed is greatly reduced.
The free-energy diagrams in Figure 8 are derived from the experimental potential energies in the actual F1 motor, with equilibrium constants chosen to mimic,
approximately, the behavior of F1 under a variety of nucleotide concentrations. It
seems that the F1 motor is designed to be an almost ideal ATP synthesizer working
under physiological conditions when driven by moderate torque (Figure 8g). For
ATP-driven rotation, fast operation requires a relatively low [ADP] (Figure 8a),
which is a condition adopted in in vitro studies. If the motor is to work as an ion
needs to be increased.
pump in vivo, as with VoV1-ATPase, effective KADP
d
ACKNOWLEDGMENTS
We thank members of the Kinosita lab and the former CREST Team 13 for collaboration and discussion. Critical comments by Y. Oono (University of Illinois),
E. Muneyuki (Tokyo Institute of Technology), and H. Nakanishi (Kyushu University) are particularly appreciated. This work was supported by Grants-in-Aid from
the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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The Annual Review of Biophysics and Biomolecular Structure is online at
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Figure 1 Observation of F1 rotation (65). Three β subunits (green) were attached to a glass
surface through histidine residues engineered on the N terminus. To the central γ subunit
(red), a 40-nm gold bead was attached through BSA and streptavidin (both brown).
Rotation was observed by dark-field microscopy. A larger bead can be observed under an
ordinary microscope, and a fluorescent actin filament attached to γ was observed under a
fluorescence microscope (40). The α subunit is in blue. The 40-nm bead and proteins in the
figure are approximately to scale.
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Figure 2 A crystal structure of mitochondrial F1 (15). α subunits, light-blue; β subunits, green (DELSEED motif in yellow at upper left); γ subunit, orange. Of the three
catalytic sites, each being at an α-β interface but hosted mainly by the β, two bind
MgADP and the third is empty. In another, quite similar structure (1), one β carries
an ATP analog instead of ADP, and thus that β is designated β-ATP. Top: Three vertical sections rotated by 120° with respect to each other. Bottom: Corresponding horizontal sections between gold lines at top. Black lines at top and black dots at bottom represent the putative axis of γ rotation (57).
Figure 5 How an ATP-driven molecular machine may work. An ATPase protein
(shown in green) does work by changing its conformation. In this example, the protein bends when it binds ATP and unbends upon release of hydrolysis products.
Brown arrows indicate how free energy decreases, accompanying work output, during the reaction. Major work is done in the process of ATP binding (and product
release), not much in conjunction with hydrolysis. Violet dots represent bonds
between ATP and the protein. Light-colored ATP on the left represents the state after
initial binding. The ATP pulls and bends the protein by progressively making more
bonds while the ATP is bound more tightly by the protein.
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Figure 6 A camshaft model for the F1 motor. Red oblong represents ATP, which is
split into larger (ADP) and smaller (Pi) orange pieces. (a) The cam (γ) is supposed
to stay in contact with the upper tip of βs, as if a thin oil layer exists at the interfaces.
(b) The basic driving force comes from a push by a β binding ATP (β2 in e1–e10),
and (c) a pull by a β from which ADP and Pi leave (β1 in e13–e16). The shade of
blue reflects the degree of bending of β. (d) In addition, the upper tip of β is assumed
to be twisted upon ATP hydrolysis, such that the tip motion produces a small amount
of counterclockwise rotation (e10–e14). Conversely, a β with its tip twisted to the
right (facing γ) has a higher affinity for ADP, whereas a tip twisted to the left favors
ATP. (e) Sequence of events during hydrolysis. When γ is forced to rotate clockwise
by an external torque (reversed sequence), the twist of the tip to the right in e16–e13
helps preferential binding of ADP over ATP.
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Figure 7 Potential energies for the camshaft model. T, ATP; H, hydrolysis products
(ADP 1 Pi); D, ADP; P, Pi; E, empty. (a) Potential diagrams for the interaction
between γ and an individual β (bottom), and the diagram for a major reaction pathway (top). (b) Diagrams showing all possible intermediates accessible from TEE
(bottom) and TTE (top). Except for the three sets of curves at the bottom of (a),
curves in a set are shifted vertically such that they represent angle-dependent free
energy of the motor and environment, as explained in text.
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Figure 8 Potential diagrams under different ∆G and external torque. In (a)–(i), nucleotide
concentrations are shown at the top and torque is shown at the bottom. The potential energy for the external torque is shown in a green line and has been added to all free-energy
curves.
Figure 9 Designing a molecular machine. (a) A portion of free-energy diagram (around
0°–120°) for the camshaft model showing TEE (purple) to ETE (blue). ∆G is the same as
ATP
in Figure 8a, but [ATP] (or KATP
a0 ) is higher and [ADP] (or Kd0 ) is lower. Thus, ATP binding is downhill (downward arrow), whereas ADP release is uphill (upward arrow).
(b) Parabolic potential energies present a high barrier in the presence of an external torque
(dotted lines).